Problem 17 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A line makes the same angle \(\theta\), with each of the \(x\) and \(z\) axis. If the angle \(\beta\), which it makes with \(y\)-axis, is such that \(\sin ^{2} \beta=3 \sin ^{2} \theta\), then \(\cos ^{2} \theta\) equals \(\quad\) [2004] (a) \(\frac{2}{5}\) (b) \(\frac{1}{5}\) (c) \(\frac{3}{5}\) (d) \(\frac{2}{3}\)

Step-by-Step Solution

Answer

\(\cos^2 \theta = \frac{3}{5}\)

Step 1: Understand the Direction Cosines

The direction cosines of a line are generally denoted by \(l\), \(m\), and \(n\) for the angles it makes with the \(x\), \(y\), and \(z\)-axes, respectively. Given the line makes the same angle \(\theta\) with the \(x\) and \(z\) axes, we have \(l = \cos \theta\) and \(n = \cos \theta\).

Step 2: Express Direction Cosine with Y-axis

We are given \(\beta\) is the angle the line makes with the \(y\)-axis, so its direction cosine is \(m = \cos \beta\).

Step 3: Use the Given Equation

We are given \(\sin^2 \beta = 3 \sin^2 \theta\). We know that \(\sin^2 \beta = 1 - \cos^2 \beta = 1 - m^2\) and \(\sin^2 \theta = 1 - \cos^2 \theta\). Substitute these into the given equation: \(1 - m^2 = 3(1 - \cos^2 \theta)\).

Step 4: Simplify the Expression

Re-arrane the equation to isolate terms involving \(m\): \(1 - m^2 = 3 - 3 \cos^2 \theta\). Simplifying gives us \(m^2 = 3 \cos^2 \theta - 2\).

Step 5: Apply Direction Cosines Property

The property of direction cosines states that \(l^2 + m^2 + n^2 = 1\). Substituting \(l = n = \cos \theta\), and the expression for \(m^2\) from the previous step, we have: \(\cos^2 \theta + (3 \cos^2 \theta - 2) + \cos^2 \theta = 1\).

Step 6: Solve Algebra

Simplify the equation: \(5 \cos^2 \theta - 2 = 1\). Solving gives \(5 \cos^2 \theta = 3\) so \(\cos^2 \theta = \frac{3}{5}\).

Step 7: Conclusion

By solving the above steps, we find that \(\cos^2 \theta = \frac{3}{5}\).

Key Concepts

Angles with Axes

When a line makes an angle with the axes in a 3-dimensional coordinate system, each of these angles helps in determining the line's orientation. Specifically, for our problem, the line makes an angle \(\theta\) with the \(x\)-axis and the \(z\)-axis, meaning the direction cosine with respect to these axes is \(\cos \theta\). The direction cosines, denoted as \(l\), \(m\), and \(n\), represent the cosine of the angles which the line makes with the \(x\)-, \(y\)-, and \(z\)-axes, respectively. For any line in space, these cosines are essential in understanding its direction. In mathematical terms, if a line makes equal angles with two axes, their respective direction cosines are equal. This basic understanding helps us link the angles involved and work towards finding unknown values through equations.

Trigonometric Identities

Trigonometric identities are crucial tools that simplify the relationships between different angles. In our problem, we know that \(\sin^2 \beta = 3 \sin^2 \theta\). Using the identity \(\sin^2 \theta = 1 - \cos^2 \theta\), we can express \(\beta\) in terms of \(\theta\). This expression allows us to substitute known values and simplify the equation further. By substituting \(\sin^2 \beta\) as \(1 - \cos^2 \beta\) and applying it to the equation, we transform the trigonometric proportion into an algebraic equation, which is easier to work with. Trigonometric identities thus bridge the gap between geometric concepts and algebraic solutions, facilitating the tractable calculation of unknown variables.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying equations to isolate and solve for unknown variables. Starting with our given equation \(1 - m^2 = 3(1 - \cos^2 \theta)\), we aim to express all terms in known forms. The step involves expanding and collecting like terms, eventually leading to \(m^2 = 3 \cos^2 \theta - 2\). From this point, algebraic manipulation guides us to substitute these back into the direction cosines' equation \(l^2 + m^2 + n^2 = 1\). Replacing \(l\) and \(n\) with \(\cos \theta\), since they are equal, allows us to solve for \(\cos^2 \theta\). The elegance of algebraic manipulation lies in its ability to transform complex relationships into coherent solutions through careful application of basic operations, culminating in the desired result, \(\cos^2 \theta = \frac{3}{5}\).